SOLUTION: 4x − 11x^1/2 + 6 = 0 Showing all necessary working, solve the equation

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Question 1152545: 4x − 11x^1/2 + 6 = 0
Showing all necessary working, solve the equation
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.

Introduce new variable  y = sqrt%28x%29.


Then your equation takes the form


    4y^2 - 11y + 6 = 0.


Solve this quadratic equation using the quadrtic formula


    y%5B1%2C2%5D = %2811+%2B-+sqrt%28%28-11%29%5E2+-+4%2A4%2A6%29%29%2F%282%2A4%29 = %2811+%2B-+sqrt%2825%29%29%2F8 = %2811+%2B-+5%29%2F8.



Case 1.  y%5B1%5D = %2811%2B5%29%2F8 = 16%2F8 = 2.


         Then  sqrt%28x%29 = 2;  hence,  x = 4.



Case 2.  y%5B2%5D = %2811-5%29%2F8 = 6%2F8 = 3%2F4.


         Then  sqrt%28x%29 = 3%2F4;  hence,  x = 9%2F16.


ANSWER.  The given equation has two and only two solutions:  x= 4  and  x= 9%2F16.


Solved, answered and explained in most detailed way, as requested. And completed.

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Introducing new variable is a standard way of solving such equations.